The native speaker told the second sentence is incorrect. Could you please explain why?
(1) What time is it?
(2) I don't know what time it is.
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First thing to note: the second sentence is not incorrect. It is correct. This answer explains why.
Let us start with normal sentence order in English. The prototypical sentence contains a verb, a subject, and an object. These typically appear in the order subject-verb-object, and English is called an SVO language for this reason. You would use that order in this instance when you say what time it is:
"It is three o'clock"
It is the subject, the verb is is (third person singular of the verb to be), and the object is three o'clock. This is the same construction as "I eat meat" or "You pet the dog".
(Some verbs can take adjectives as objects; to be always can, as in "you are pretty", and any instance where you could replace the verb with to be and it make sense, clearly related to the original sentence, can also take an adjective object. These are called linking verbs, and other than to be they are generally describing perception, like look or seem, or change, like grow or become. That's not relevant to this question, but it's worth throwing in as an aside.)
When we form questions out of simple sentences, though, word order changes. The simplest way of doing this is with an interrogative, which is usually a "wh-question" word, like who, how, what and so on, or an appropriate conjugation of an auxiliary verb like to be, to do, etc. The interrogative goes at the start of the sentence, and then the other three elements come after it in a range of different orders. Where the sentence originally used a verb that can be auxiliary as its verb, it doesn't need repeating, so the question version of "you are hungry" is "are you hungry?" This is simple subject-verb inversion, which you see in a lot of questions with to be: "is it red?", "am I boring?". It used to be common, if we can judge from old text, with other verbs as well, with examples like "go you home?" or even with more complex things involving verbs that are not generally used as an auxiliary, like "think you it goes well?", which we would now render as "do you think it's going well?" Where a question starts with an auxiliary verb, it is generally a yes/no question.
"Wh-question" interrogatives behave a little more interestingly, as they can effectively become the object of the sentence. If you know what someone wants to wear, you might say "you want to wear a skirt". You can even make that a question as it is, though it would be unusual unless you wanted to express incredulity or scepticism or such; generally a simple statement intoned as a question will be taken in that way, though there's also the case of an observation intoned as a question as a way of seeking confirmation. For example, if someone comes in from outside and is soaking wet, you might ask "is it raining?".
If you don't know what someone wants to wear, you ask "what do you want to wear?". The "do" is an extra auxiliary that is often necessary (though it could be some other auxiliary), though if there's a noun specifying what category of answer you want, or forming part of the object, it isn't; "what colour", "which perfume" and so on. If you don't know what something is, you say "what is that?", but if you know what it is, you might say "that is an elephant". Sometimes, non-native speakers who are still learning might ask "that is what?", which shows the question in SVO order. If you are specifying that you are enquiring after an animal, it would be "what animal is that?", or after a colour, "what colour is that?" (or "which colour is that?", as colour is often treated as as choice from a selection).
So, when saying the time, you say "it is three o'clock", subject-verb-object. Maintaining that order, you would try to ask the question "it is what time?", and people would know what you meant but know you were far from fluent in English. Because it's is that sort of interrogative, the order shifts to "what time is it?".
So, that explains your first example, and you can see that it is correct. The second example is also correct, but we haven't seen why yet. By pulling the sentence apart, we can see that I is the subject, the principle verb is to know, we have an auxiliary providing negation with don't, and finally the object is what time it is, a phrase which we can then break down - but not as a sentence. It doesn't follow the rules to be a sentence on its own, and where it fits into the wider sentence means you would never expect it to. It is taking the place of a noun - the object - in the sentence, so it is a noun phrase.
Noun phrases have rules all of their own. They usually have to start with a determiner, which can be a number or article, but it can also be a wh-question word. When you create a fairly simple noun phrase with a verb and a wh-question word, it usually takes the form seen here - determiner-subject-verb. Actually, you can look at "what time" in "what time is it?" as a noun phrase as well, but interrogatives have separate rules of their own, as touched on above, so we don't usually think of them as such. Sentences asking questions behave differently, but "I don't know what time it is" is not a question. "what time it is" in that sentence is a noun phrase, and follows the rules for a noun phrase.
Noun phrases can be simpler than that; an adjective and a noun together can be a noun phrase, and then they don't need a determiner. Because that sort is that simple, discussions of noun phrases often ignore them. They can also be much more complex, such as "the young lady wearing the green dress, at the back of the room". But the reason "what time it is" in example two has the word order it does is because of the type of noun phrase it is.